are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of i.i.d. random variables conditioned on the number of these variables ():

is a well-defined distribution. In the case N = 0, then the value of Y is 0, so that then Y | N = 0 has a degenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, where this joint distribution is obtained by combining the conditional distribution Y | N with the marginal distribution of N.

For more special case of DCP, see the reviews paper[7] and references therein. For example, the Luria–Delbrück distribution in Luria–Delbrück experiment.

This distribution can model batch arrivals (such as in a bulk queue.[4][8] The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.[3]

When some are non-negative, it is the discrete pseudo compound Poisson distribution.[3]

If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the "compound Poisson distribution" outlined above is essentially the same as the more general class of compound probability distributions. However, the properties outlined above do depend on its formulation as the sum of a Poisson-distributed number of random variables. The distribution of Y in the case of the compound Poisson distribution with exponentially-distributed summands can be written in an form.[9][10]

where the sum is by convention equal to zero as long as N(t)=0. Here, is a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of [11]

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.[12]

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim[9] to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which has an exponential distribution. Thompson[10] applied the same model to monthly total rainfalls.